Rational Cuspidal Curves in a moving family of $\mathbb{P}^2$

TL;DR

This paper derives a formula for counting rational degree d curves with a cusp in a moving family within projective space, extending classical enumerative geometry results to a family setting.

Contribution

It introduces a new method to compute the number of rational cuspidal curves in a family of projective planes, generalizing previous fixed-configuration counts.

Findings

Derived a formula for the number of rational cuspidal curves in a family of $P^2$

Validated results against known counts for planar cubics and quartics

Extended classical enumerative geometry to a family context

Abstract

In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in $\mathbb{P}^2$, which has been studied earlier by Z. Ran, R. Pandharipande and A. Zinger. We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger and I. Biswas, S. D'Mello, R. Mukherjee and V. Pingali. We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author, where they compute the characteristic number of $\delta$-nodal planar curves in $\mathbb{P}^3$ with one cusp (for $\delta \leq 2$).